arXiv 1702.026672017-02math.DG

关于里奇流的存在性和曲率估计

On existence and curvature estimates of Ricci flow

Man-Chun Lee, Luen-Fai Tam

备注：: This is an expanded version of the previous paper "A note on existence of Ricci flow"

摘要：

本文利用He的方法证明了完备非紧黎曼流形上里奇流的短时存在性，并证明了它具有以下性质：（i）存在$r_0>0$使得半径为$r\le r_0$的测地球的体积接近于欧氏空间中半径相同的测地球的体积;（ii）Ricci曲率有界;（iii）截面曲率以$-Cd^2(x,x_0)$为界，其中$d(x,x_0)$是到固定点的距离。给出了具有非负对分曲率和最大体积增长的完备非紧Kahler流形的一致化的一个应用。这个结果与He和Liu的结果有关。证明了如果$g(t)$是完备非紧Kahler流形上Kahler-里奇流的完备解，使得$g(0)$具有非负的对分曲率，且对某些$a>0, 2>b>0$，$g(t)$的曲率由$at^{-b}$有界，则$g(t)$也具有非负的对分曲率.这推广了以前的一个结果。在$g(0)$是非塌缩的假设下，并在$g(t)$的曲率一致有界的条件下，证明了对于某些$a>0$，$g(t)$的曲率实际上是由$at^{-1}$有界的.这一结果是证明使用的方法由西蒙和托普。

英文摘要：

In this work, using the method by He, we prove a short time existence for Ricci flow on a complete noncompact Riemannian manifold with the following properties: (i) there is $r_0>0$ such that the volume of any geodesic balls of radius $r\le r_0$ is close to the volume of the geodesic ball in the Euclidean space with the same radius; (ii) the Ricci curvature is bounded below; (iii) the sectional curvature is bounded below by $-Cd^2(x,x_0)$ where $d(x,x_0)$ is the distance from a fixed point. An application to the uniformization for complete noncompact Kahler manifolds with nonnegative bisectional curvature and maximal volume growth is given. The result is related to the results of He and Liu. We prove if $g(t)$ is a complete solution to the Kahler-Ricci flow on a complete noncompact Kahler manifold so that $g(0)$ has nonnegative bisectional curvature and the curvature of $g(t)$ is bounded by $at^{-b}$ for some $a>0, 2>b>0$, then $g(t)$ also has nonnegative bisectional curvature. This generalizes a previous result. Under the assumption that $g(0)$ is noncollapsing, with additional assumption that the curvature of $g(t)$ is uniformly bounded away from $t>0$, we prove that the curvature of $g(t)$ is in fact bounded by $at^{-1}$ for some $a>0$. This result is proved using the method by Simon and Topping.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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